Mr. Reider AP Stat November 18, 2010 Sections 6.1 and 6.2 Mr. Reider AP Stat November 18, 2010

Drill What is probability? What does it mean to be random?

Section 6.1 Randomness

Definitions We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of time the outcome would occur in a very long series of repetitions.

Probability Model A probability model consist of two parts: A sample space A way of assigning probabilities to events

Section 6.2 Probability Models

Sample Space The Sample Space (denoted S) of a random phenomenon is the set of all possible outcomes.

What is the Sample Space of… Flipping a coin? Rolling a die? Drawing a card from a standard deck?

Multiplication Principle If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in a ∙ b number of ways. What if we are doing three tasks? Four tasks? Tree diagrams

Example Flipping a coin and rolling a die

Simple Set Theory Intersection: Denoted Read “A and B” Union: Denoted Read “A or B”

Events An event is an outcome, or set of outcomes, of a sample space. An event, E, is a subset S. What does it mean to be a subset?

Events The complement of any event is that A does not occur. It is denoted AC.

Events Two events are considered disjoint if they have no common outcomes. Note: “disjoint” and “mutually exclusive” are synonymous

Notation

Probability Rules Let A and B be events Addition Rule: If A and B are disjoint, then

Probability in a Finite Sample Space If there are a finite set of outcomes in S, we can assign a probability to each individual outcome. What conditions do we need to meet with these probabilities?

Example What is P(Married)? Marital status: Never married Married Widowed Divorced Probability: .298 .622 .005 .075 What is P(Married)? P(Married) =.622

Example What is P(not Married)? Marital status: Never married Married Widowed Divorced Probability: .298 .622 .005 .075 What is P(not Married)? P(not Married) = 1-.622 =.378 (Complement Rule)

Example What is P(Never married or Divorced)? Marital status: Widowed Divorced Probability: .298 .622 .005 .075 What is P(Never married or Divorced)? Since “Never married and Divorced are disjoint, P(Never married or Divorced) = .298+.075 =.373 (Addition Rule for disjoint events)

Equally Likely Outcomes If a random phenomenon has k possible out comes, all of them equally likely, then each individual outcome has a probability of . The probability of any event A is

Independent Events Two events are considered independent if knowing that one event occurs does not change the probability of the other occurs. What do you think are some examples of independent events? Rolling a number cube twice, flipping a coin twice, drawing two cards with replacement, winning a contest.

Multiplication Rule If A and B are independent events, then

Probability Rules Let A and B be events Addition Rule: If A and B are disjoint, then Multiplication Rule: If A and B are mutually exclusive, then

Examples A = {rolling a 2 or 5 on a die.} B = {Drawing a face card from a standard deck of playing cards} C= {Rolling a 3 or 4 and drawing an ace}

Example